Vistara will check you thru and your bags will also be sent to WAW, no issues there. Arriving in DEL you can go to the transfer desk, get the 'DT' mark on your BP, but finding an agent at 6pm for next days flight will be tough. Worst case you can write 'DT' on the BP yourself. For the location of the two 'DT' marks see this post:
https://www.flyertalk.com/forum/26663144-post13.html
Once airside there is choice of lounges, but do you really want to spend that much time in a lounge? I usually spend 3-4 hours in DEL T3 and after 2 hours it gets very very boring. There are napcabs in the concourse where you could theoretically go sleep. I would enter India, stay at a hotel in aerocity (short 10 min cab ride away) and come back the next morning.